*24. THE UNIVERSAL CLASS, THE NULL-CLASS, AND THE EXISTENCE OF CLASSES.
Project Gutenberg EPUB3 edition
Whitehead, Alfred North and Russell, Bertrand
Summary of *24.
The universal class, denoted by , is the class of all objects of the type which, in the given context, is being denoted by small Latin letters, i.e. of the lowest type concerned. Thus , like "," is ambiguous as to type. Its definition is as follows:
*24·01.
Any other property possessed by everything would do as well as "," but this is the only such property which we have hitherto studied.
The null-class, denoted by , is the class which has no members. Like , it is ambiguous as to type. We use the same symbol, , for null-classes of various types; but these null-classes differ. The type of is determined by that of the terms concerning which "" is false: whatever may be, "" will not represent a true proposition, but unless is of the appropriate type, "" will be meaningless, not false. Thus is of the type next above that of an concerning which "" is significant and false. The definition of is
*24·02.
When a class is not null, so that it has one or more members, it is said to exist. (This sense of "existence" must not be confused with that defined in *14·02.) We write "" for " exists." The definition is
*24·03.
In the present number, we shall deal first with the properties of and , then with those of existence. In comparing the algebra of symbolic logic with ordinary algebra, takes the place of , while combines the properties of and of .
Among the more important properties of and which are proved in this number are the following:
*24·1.
I.e. "nothing is not everything." This is useful as giving us the existence of at least two classes. If the monistic philosophers were right in maintaining that only one individual exists, there would be only two classes, and , being (in that case) the class whose only member is the one individual. Our primitive propositions do not require the existence of more than one individual.
*24·102·103 show that any function which is always true determines the universal class, and any function which is always false determines the null-class.
*24·21·22 give forms of the laws of contradiction and excluded middle, namely "nothing is both and not-" ( and "everything is either or not-" (.
*24·23·24·26·27 give the properties of and with respect to addition and multiplication, namely: multiplication by and addition of make no change in a class (*24·26·24); addition of gives , and multiplication by gives (*24·27·23). It will be observed that the properties of and result from each other by interchanging addition and multiplication.
*24·3.
I.e. " is contained in " is equivalent to "nothing is but not ."
*24·311.
I.e. "no is a " is equivalent to "nothing is both and ."
*24·411.
*24·43.
As a rule, propositions concerning are much less used than the correlative propositions concerning .
The properties of the existence of classes result from those of , owing to the fact that is the contradictory of , as is proved in *24·54. Thus we have, in virtue of *24·3,
*24·55.
I.e. "not all 's are 's" is equivalent to "there are 's which are not 's." This is the familiar proposition of formal logic, that the contradictory of the universal affirmative is the particular negative.
We have
*24·56.
*24·561.
I.e. if a sum exists, then one of the summands exists, and vice versa; and if a product exists, both the factors exist (but not vice versa).
The proofs of propositions in the present number offer no difficulty.
*24·01.
*24·02.
*24·03.
*24·1.
*24·101.
*24·102.
Dem.
Thus any function which is always true determines the universal class, and vice versa.
*24·103.
Dem.